Heat olive oil in a large pot over medium heat. Add green onion and garlic and saut? for 30 to 60 seconds until fragrant.

Pour in chicken broth and bring to a boil over medium-high heat.

Stir in rice, oregano, bay leaves, and season with salt and pepper to taste. Reduce heat to medium-low. Cover and simmer, stirring occasionally, until rice is just tender, about 15 minutes for jasmine rice and 20 minutes for white rice.

Near the end of cooking, whisk together eggs, egg yolks, and lemon juice in a medium mixing bowl until frothy for about 1 minute. Slowly pour in 1 cup hot broth from soup while whisking.

Stir in chicken and lemon zest. Remove from heat and while stirring soup, pour in egg mixture. Return to heat and cook over low heat, stirring constantly, until soup has thickened slightly for a minute or two.

Serve with parsley and/or dill. Add more chicken broth to thin soup if desired.